Example Data Table
Use this sample system to test the calculator. It gives x1 = 2, x2 = 3, and x3 = -1.
| Equation |
a1 |
a2 |
a3 |
b |
Meaning |
| 1 |
2 |
1 |
-1 |
8 |
2x1 + x2 - x3 = 8 |
| 2 |
-3 |
-1 |
2 |
-11 |
-3x1 - x2 + 2x3 = -11 |
| 3 |
-2 |
1 |
2 |
-3 |
-2x1 + x2 + 2x3 = -3 |
Formula Used
The calculator solves a square linear system written as Ax = b. It creates the augmented matrix [A|b]. Then it applies row operations until the coefficient part becomes upper triangular.
For a pivot in row k, every lower row i is updated with this rule:
R_i = R_i - (a_ik / a_kk) R_k
After elimination, back substitution starts from the last row:
x_i = (b_i - a_i,i+1 x_i+1 - ... - a_i,n x_n) / a_i,i
The determinant is the product of the final diagonal entries. Its sign changes after every row swap. Rank checks compare the coefficient matrix with the augmented matrix.
How to Use This Calculator
1. Select size.
Choose the number of equations and variables.
2. Enter coefficients.
Fill each coefficient and right-side constant.
3. Choose pivoting.
Partial pivoting is a safe default.
4. Set precision.
Use more decimals for sensitive systems.
5. Calculate.
Read the solution and row operations.
6. Export.
Download CSV or PDF for records.
Understanding the Gauss Elimination Method
Gauss elimination is a direct method for solving systems of linear equations. It changes a system into an easier equivalent system. The method keeps the same solution while reshaping the matrix. Each row operation represents a valid equation change. This makes it useful for algebra, engineering, economics, and numerical analysis.
Why Pivoting Matters
A pivot is the number used to remove entries below it. A small pivot can create large rounding errors. Partial pivoting swaps rows so the strongest available pivot is used. Scaled partial pivoting also considers row size. This is helpful when rows contain very different magnitudes. The calculator offers both choices, plus a no-pivot option for classroom comparison.
What the Steps Show
The displayed steps show every row swap and elimination operation. You can follow each matrix state from the original system to upper triangular form. This is useful when checking homework. It also helps teachers explain where each number comes from. After the forward phase, the calculator performs back substitution. The values of the unknowns are then listed clearly.
Rank, Determinant, and Residuals
Advanced checks help explain the type of answer. If the rank of the coefficient matrix is less than the augmented rank, the system is inconsistent. It has no solution. If the coefficient rank is less than the number of variables, free variables exist. The system has infinitely many solutions. For a unique square system, a nonzero determinant confirms invertibility. Residual values check the answer by inserting it back into the original equations.
Practical Use
This tool is best for small and medium systems where transparent work is important. It is not only a final answer generator. It is a learning aid and a verification tool. The CSV export is useful for spreadsheets. The PDF export is useful for assignments, reports, or saved documentation. Always review tolerance and precision when numbers are very small.
FAQs
1. What does this calculator solve?
It solves square systems of linear equations using Gaussian elimination. Enter coefficients and constants, then review solutions, ranks, determinant, residuals, RREF, and row operations.
2. What is partial pivoting?
Partial pivoting swaps rows to place a larger pivot in the active row. It usually improves numerical stability and reduces rounding problems during elimination.
3. When should I use scaled pivoting?
Use scaled pivoting when rows have very different coefficient sizes. It compares pivot strength relative to each row, which can improve reliability for uneven data.
4. What does zero tolerance mean?
Zero tolerance decides when a tiny number should be treated as zero. A smaller value is stricter. A larger value can help ignore rounding noise.
5. Why does the calculator show no solution?
No solution appears when the augmented matrix has higher rank than the coefficient matrix. This means the equations contradict each other.
6. Why are there infinitely many solutions?
Infinitely many solutions occur when the coefficient rank is lower than the number of variables, but the system is still consistent. Free variables remain.
7. What is the residual check?
The residual checks Ax - b after the solution is found. Values near zero mean the computed solution fits the original equations well.
8. Can I export the work?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a readable report with results and visible step information.